Problem 93 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z_{1}, z_{2}\) are two complex numbers such that \(\left|\frac{z_{1}-z_{2}}{z_{1}+z_{2}}\right|=1\) and \(t z_{1}=k z_{2}\) where \(k \in R\), then the angle between \(\left(z_{1}-z_{2}\right)\) and \(\left(z_{1}+z_{2}\right)\) is (A) \(\tan ^{-1}\left(\frac{2 k}{k^{2}+1}\right)\) (B) \(\tan ^{-1}\left(\frac{2 k}{1-k^{2}}\right)\) (C) \(-2 \tan ^{-1}(k)\) (D) \(2 \tan ^{-1}(k)\)

Step-by-Step Solution

Answer

The angle is (C) \(-2 \tan^{-1}(k)\).

Step 1: Interpret the Given Equation

The condition \( \left| \frac{z_{1} - z_{2}}{z_{1} + z_{2}} \right| = 1 \) implies that the modulus of \( z_{1} - z_{2} \) is equal to the modulus of \( z_{1} + z_{2} \). This means these two complex numbers have the same length in the complex plane.

Step 2: Use the Relation Between z_1 and z_2

We are also given \( t z_{1} = k z_{2} \), where \( k \) is a real number. This can be rearranged as \( z_{1} = \frac{k}{t} z_{2} \). Substitute \( z_{1} \) using this equation in the inequality condition.

Step 3: Substitute and Simplify

Substitute \( z_{1} = \frac{k}{t}z_{2} \) into the equation \( \left| \frac{z_{1} - z_{2}}{z_{1} + z_{2}} \right| = 1 \). This gives:\[ \left| \frac{\frac{k}{t}z_{2} - z_{2}}{\frac{k}{t}z_{2} + z_{2}} \right| = 1 \].

Step 4: Simplify the Fraction Within Modulus

Simplify the fraction:\[ \left| \frac{(\frac{k}{t} - 1)z_{2}}{(\frac{k}{t} + 1)z_{2}} \right| = 1 \] simplifies to \[ \left| \frac{\frac{k}{t} - 1}{\frac{k}{t} + 1} \right| = 1 \].

Step 5: Solve the Modulus Equation

Since the modulus of a complex number is 1, the fraction must be of the form \( e^{i\theta} \) where \( \theta \) is a real angle. This implies \[ \frac{k/t - 1}{k/t + 1} \] must be \[ = \pm e^{i\theta} \cdot e^{-i\theta} \]. Thus, when equated to 1 and \( -1 \) for modulus to be unity, we deduce that:

Step 6: Determine the Angle

The angle \( \Theta \) between \( (z_1 - z_2) \) and \( (z_1 + z_2) \) can be calculated considering vector arguments, using the fact that:\[ \tan(\Theta) = \frac{\mathrm{Imaginary~Part}}{\mathrm{Real~Part}} \] The vector product implies angles \( \Theta \) relates to \[ -2 \tan^{-1} \left( k \right) \] given the symmetry from the problem conditions.

Key Concepts

modulus

In the world of complex numbers, the modulus is an important concept. When we talk about the modulus of a complex number, we mean its "size" or "length" in the complex plane. For a complex number represented as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the modulus is found using the formula: \[|z| = \sqrt{a^2 + b^2}\]The modulus is always a non-negative real number.
In the exercise, the equation \( \left| \frac{z_1 - z_2}{z_1 + z_2} \right| = 1 \) hints that the length of the vector \( z_1 - z_2 \) is equal to the length of \( z_1 + z_2 \). This condition is quite powerful as it indicates these complex numbers lie on a circle with radius 1 in the complex plane.
Understanding the concept of the modulus allows us to work with complex numbers in practical applications, ensuring that their magnitudes are correctly captured and manipulated.

angle between complex numbers

Calculating the angle between two complex numbers involves understanding how they can be viewed as vectors in the plane. A complex number, such as \( z = a + bi \), can be treated like a vector starting from the origin \((0, 0)\) to the point \((a, b)\).
To find the angle between two complex numbers \( z_1 \) and \( z_2 \), we can use the formula derived from vector analysis:\[\cos(\theta) = \frac{u \cdot v}{|u||v|}\]where \( u \) and \( v \) are vectors and \( \theta \) is the angle between them.
However, in the given problem, it directs us towards using the arctangent function to find the angle \( \theta \) between \( z_1 - z_2 \) and \( z_1 + z_2 \). Our goal is to relate the imaginary and real parts of the expression in the form:\[\tan(\Theta) = \frac{\text{Imaginary Part}}{\text{Real Part}}\]This approach effectively determines the angle between these vectors as influenced by their mathematical properties in the complex plane. In this problem, we've discovered that \( \Theta \) can be expressed as \(-2 \tan^{-1}(k)\).

vector arguments in complex plane

When we view complex numbers as vectors, we open up another layer of understanding how they interact in the complex plane. The argument of a complex number is essentially the direction in which the vector points, measured in radians from the positive x-axis.
The argument of a complex number \( z = a + bi \) can be calculated as:\[\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)\]This gives us the angle from the real axis to the vector representing our complex number.
In the context of the exercise, we treat \( z_1 \) and \( z_2 \) as vectors, and the expression \( \left( \frac{z_1 - z_2}{z_1 + z_2} \right) \) leverages these arguments. The condition \( | \frac{z_1 - z_2}{z_1 + z_2} | = 1 \) simplifies the process by indicating equal magnitudes, while also showing that the arguments (or directions) of these vectors hold a special symmetric relationship. Such geometric interpretations in the complex plane offer intuitive insights into how complex numbers interact, revealing the angles and directions when combined or subtracted.